ECON 707/807: Econometrics II

Time Series

Sequential measurements or values of a single entity over time.

Why study time series data?

Gross Domestic Product
Unemployment
Vehicle Demand
Energy Consumption

Forecasting

“Predict” outcomes of a time series in future (unobserved) periods.

Time Series Models

\[Y_{t} = f(Y_{t-1}, X_{t}, X_{t-1}, \tau_t, C_t, S_t)\] where:

\(Y_{t-1}\) is a lagged value of \(Y\)
\(X_t\) is a contemporaneous independent value
\(X_{t-1}\) is a lagged value of \(X\)
\(\tau\) is the trend
\(C\) is the cycle
\(S\) is the season

Statistics Review

mean
variance
standard error
central limit theorem (CLT)
p-value
\(Y_i = \alpha + \beta X_i + \epsilon_{i}\)
causality
spurious

Forecasting Steps

Define Problem

What am I trying to solve?
Gather Data

FRED, WRDS, Etc.
Exploratory Data Analysis (EDA)

Plot, Plot, Plot
Choose & Fit a Model

\(Y_{t} = f(Y_{t-1}, X_{t}, X_{t-1}, \tau_t, C_t, S_t)\)
Evaluate, Forecast

Packages and Functions

table()
aggregate()
match()
paste(), paste0(), substr(), grepl(), gsub(), regexpr(), strsplit(), unlist()

Packages and Functions

for()
if(), else
ifelse()
fixest
scales::alpha()
data.table::fread()
readRDS(), saveRDS()

Packages and Functions

lubridate¹
tsibble
forecast
ts()²
duplicated()

Install and Load Packages

Code

install.packages("lubridate")
library("lubridate")

Make a Time Series

Code

urate <- read.csv("../data/unrate_us.csv")
colnames(urate) <- c("date", "urate_t")
head(urate)

        date urate_t
1 1948-01-01     3.4
2 1948-02-01     3.8
3 1948-03-01     4.0
4 1948-04-01     3.9
5 1948-05-01     3.5
6 1948-06-01     3.6

Plot a Time Series

Code

urate$date <- ymd(urate$date)
plot(urate$date, urate$urate_t,
     xlab = "Month",
     ylab = "Unemployment Rate",
     main = "Monthly Unemployment Rate")

Plot a Time Series

Code

plot(urate$date, urate$urate_t,
     ylab = "Unemployment Rate",
     xlab = "Time",
     main = "Monthly Unemployment Rate",
     type = "l", col = "dodgerblue")

Lags (Leads)

        date urate_t
1 1948-01-01     3.4
2 1948-02-01     3.8
3 1948-03-01     4.0
4 1948-04-01     3.9
5 1948-05-01     3.5
6 1948-06-01     3.6

Code

t <- 2:nrow(urate)
urate$urate_tm1 <- NA
urate$urate_tm1[t] <- urate$urate_t[t-1]
urate$urate_tp1 <- NA
urate$urate_tp1[t-1] <- urate$urate_t[t]
head(urate)

        date urate_t urate_tm1 urate_tp1
1 1948-01-01     3.4        NA       3.8
2 1948-02-01     3.8       3.4       4.0
3 1948-03-01     4.0       3.8       3.9
4 1948-04-01     3.9       4.0       3.5
5 1948-05-01     3.5       3.9       3.6
6 1948-06-01     3.6       3.5       3.6

What about panels?

Code

df <- data.frame(state = c(rep("NY", 10),
                           rep("VA", 10),
                           rep("CA", 10)),
                 year = rep(2010:2019, 3),
                 var = rnorm(30))
df[df$year %in% 2010:2012,]

   state year         var
1     NY 2010  0.98176529
2     NY 2011 -0.04575189
3     NY 2012  1.73884110
11    VA 2010 -1.23379611
12    VA 2011 -0.97220151
13    VA 2012 -1.20317155
21    CA 2010  0.06060990
22    CA 2011  0.68533755
23    CA 2012  0.48815305

What about panels?

Code

df$var_lag1 <- NA

   state year         var var_lag1
1     NY 2010  0.98176529       NA
2     NY 2011 -0.04575189       NA
3     NY 2012  1.73884110       NA
11    VA 2010 -1.23379611       NA
12    VA 2011 -0.97220151       NA
13    VA 2012 -1.20317155       NA
21    CA 2010  0.06060990       NA
22    CA 2011  0.68533755       NA
23    CA 2012  0.48815305       NA

What about panels?

Code

df$var_lag1 <- NA
t <- which(duplicated(df$state))
df$var_lag1[t] <- df$var[t-1]
df[df$year %in% 2010:2012,]

   state year         var    var_lag1
1     NY 2010  0.98176529          NA
2     NY 2011 -0.04575189  0.98176529
3     NY 2012  1.73884110 -0.04575189
11    VA 2010 -1.23379611          NA
12    VA 2011 -0.97220151 -1.23379611
13    VA 2012 -1.20317155 -0.97220151
21    CA 2010  0.06060990          NA
22    CA 2011  0.68533755  0.06060990
23    CA 2012  0.48815305  0.68533755

Multiple Plots

Code

plot(df$year, df$var)

Multiple Plots

Code

plot(df$year, df$var,
     col = match(df$state, unique(df$state)))

Multiple Plots

Code

plot(df$year, df$var,
     col = match(df$state, unique(df$state)),
     type = "l")

Multiple Plots

Code

lim <- df$state == "NY"
plot(df$year[lim], df$var[lim],
     col = match(df$state[lim], unique(df$state)),
     type = "l")

Multiple Plots

Code

lim <- df$state == "NY"
plot(df$year[lim], df$var[lim],
     col = match(df$state[lim], unique(df$state)),
     type = "l")
lim <- df$state == "VA"
lines(df$year[lim], df$var[lim],
     col = match(df$state[lim], unique(df$state)))
lim <- df$state == "CA"
lines(df$year[lim], df$var[lim],
     col = match(df$state[lim], unique(df$state)))

Multiple Plots

Code

plot(df$year, df$var, type = "n")
lim <- df$state == "NY"
lines(df$year[lim], df$var[lim],
     col = match(df$state[lim], unique(df$state)))
lim <- df$state == "VA"
lines(df$year[lim], df$var[lim],
     col = match(df$state[lim], unique(df$state)))
lim <- df$state == "CA"
lines(df$year[lim], df$var[lim],
     col = match(df$state[lim], unique(df$state)))

Next Class

\(E[y]\), \(\hat{y}\)
Loss Functions
Lags, Leads
Conditional Forecasts
Time Series Components

Practice - Cleaning Data

Work on #5.
Work on #8.

Practice - Social Security

Navigate to the following link: here
Download the state-specific data.
Read in the data from Virginia.
- Make some plots for a name of your choosing (time series, distributions, etc.)
Read in all of the data
- Combine into one data.frame.
- What state is your chosen name “most” popular?
- What year is your chosen name “most” popular?
- Analyze the “gender neutrality” of a relatively unisex name (e.g. “Alex”).
- Analyze a common name and it’s nickname (e.g. “Alex” vs “Alexander”).